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 A173675 Let d_1, d_2, d_3, ..., d_tau(n) be the divisors of n; a(n) = number of permutations p of d_1, d_2, d_3, ..., d_tau(n) such that p_(i+1)/p_i is a prime or 1/prime for i = 1,2,...,tau(n)-1 and p_1 <= p_tau(n). 4
 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 8, 1, 4, 4, 1, 1, 8, 1, 8, 4, 4, 1, 14, 1, 4, 1, 8, 1, 72, 1, 1, 4, 4, 4, 20, 1, 4, 4, 14, 1, 72, 1, 8, 8, 4, 1, 22, 1, 8, 4, 8, 1, 14, 4, 14, 4, 4, 1, 584, 1, 4, 8, 1, 4, 72, 1, 8, 4, 72, 1, 62, 1, 4, 8, 8, 4, 72, 1, 22, 1, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Variant of A179926 in which the permutation of the divisors may start with any divisor but the first term may not be larger than the last term. From Andrew Howroyd, Oct 26 2019: (Start) Equivalently, the number of undirected Hamiltonian paths in a graph with vertices corresponding to the divisors of n and edges connecting divisors that differ by a prime. a(n) depends only on the prime signature of n. See A295786. (End) LINKS Andrew Howroyd, Table of n, a(n) for n = 1..2048 V. Shevelev, Combinatorial minors of matrix functions and their applications, arXiv:1105.3154 [math.CO], 2011-2014. V. Shevelev, Combinatorial minors of matrix functions and their applications, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014). Robert G. Wilson v, Re: A combinatorial problem, SeqFan (Aug 02 2010) FORMULA From Andrew Howroyd, Oct 26 2019: (Start) a(p^e) = 1 for prime p. a(A002110(n)) = A284673(n). a(n) = A295786(A101296(n)). (End) EXAMPLE a(1) = 1: [1]. a(2) = 1: [1,2]. a(6) = 4: [1,2,6,3], [1,3,6,2], [2,1,3,6], [3,1,2,6]. a(12) = 8: [1,2,4,12,6,3], [1,3,6,2,4,12], [1,3,6,12,4,2], [2,1,3,6,12,4], [3,1,2,4,12,6], [3,1,2,6,12,4], [4,2,1,3,6,12], [6,3,1,2,4,12]. MAPLE with(numtheory): q:= (i, j)-> is(i/j, integer) and isprime(i/j): b:= proc(s, l) option remember; `if`(s={}, 1, add( `if`(q(l, j) or q(j, l), b(s minus{j}, j), 0), j=s)) end: a:= proc(n) option remember; ((s-> add(b(s minus {j}, j), j=s))(divisors(n)))/`if`(n>1, 2, 1) end: seq(a(n), n=1..100); # Alois P. Heinz, Nov 26 2017 MATHEMATICA b[s_, l_] := b[s, l] = If[s == {}, 1, Sum[If[PrimeQ[l/j] || PrimeQ[j/l], b[s ~Complement~ {j}, j], 0], {j, s}]]; a[n_] := a[n] = Function[s, Sum[b[s ~Complement~ {j}, j], {j, s}]][ Divisors[n]] / If[n > 1, 2, 1]; Array[a, 100] (* Jean-François Alcover, Nov 28 2017, after Alois P. Heinz *) CROSSREFS Cf. A000005, A000040, A002110, A101296, A179926, A284673, A295786. See A295557 for another version. Sequence in context: A340227 A351942 A351230 * A364360 A085731 A131301 Adjacent sequences: A173672 A173673 A173674 * A173676 A173677 A173678 KEYWORD nonn AUTHOR N. J. A. Sloane, Nov 24 2010 EXTENSIONS Alois P. Heinz corrected and clarified the definition and provided more terms. - Nov 07 2014 STATUS approved

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Last modified September 23 11:05 EDT 2023. Contains 365544 sequences. (Running on oeis4.)